耦合非线性系统的自组织及其集体动力学行为是当前非线性科学研究中的一个重要课题。在对单个非线性系统中的混沌或非混沌行为及规律逐渐熟知以后，从上世纪八十年代以来在非线性动力学方面的工作逐渐转移到耦合非线性系统的研究中，主要关心耦合系统中的协同行为，其中包括系统自组织形成的各种集体行为如同步、锁相、锁频、振动死亡、非线性波、延展态、反相态等的产生机制及其相互转化。在本文中，我们重点研究耦合非线性系统集体行为的同步、锁频、锁相等问题，以Kuramoto相振子组成的系统为研究对象，采用理论分析和数值计算相结合的方法，讨论耦合系统达到完全同步、部分同步以及相同步的状态时出现的合作现象以及动力学行为，揭示其内在规律。本文的具体结构安排如下：第一章简要阐述非线性动力学系统的基本背景知识，包括非线性系统动力学方程的表述形式、解的线性稳定性分析、定点的种类以及分岔类型。首先引入描述非线性耦合系统的常用动力学方程，即Kuramoto模型，给出了该模型在全局耦合下的平均场解和局域耦合系统的相关研究方法和成果。然后介绍了常见复杂网络的基本概念，网络类型以及网络同步的判定方法，复杂网络上的耦合 Kuramoto 振子系统的同步理论分析。了解这些基本知识将为我们后面讨论的研究工作打下基础。第二章主要讨论 Kuramoto 振子环的一种特殊同步方式，即单集团同步（SCS）。我们发现当自然频率分布满足某种特殊构型时，系统会产生单集团同步现象，而且随着系统振子数 N 的不断增加，临界耦合强度 Kc 会趋于一个有限饱和值。我们对有限大小系统的分析工作，为验证在热力学极限条件下这一结论仍然成立提供了有力依据。我们基于两种不同方法，即随机改变任意两个振子的位置和给予所有振子自然频率不同程度的随机扰动，证实产生单集团同步方式（ SCS）的构型具有鲁棒性。所有这些结果表明单集团同步方式具有一般存在性，且具有一定的潜在工程应用价值。第三章我们研究由 Kuramoto 耦合相振子组成无标度网络中出现的爆炸同步现象。爆炸同步现象可以由系统中振子的自然频率和振子的度相关引起，我们加入阻挫来分析系统同步所受的影响。研究结果表明阻挫可以促进或推迟同步，同时随着阻挫值的变化会出现去同步现象以及相变的连续性也会发生改变。为了解释 BA 网络中阻挫对系统同步行为的影响，我们研究具有典型无标度网络特性的星型网络，理论分析和数值模拟结果一致。我们还提出自洽方法来解释阻挫达到一定范围时出现去同步现象的原因。在工程实践上不连续突变具有一定的危险性，而阻挫则是一个比较好控制的连续参量，因此我们的研究结果对复杂网络的同步控制具有重要的启示作用。第四章研究一维闭合环上Kuramoto相振子在非对称的耦合作用下，在同步区域出现多定态现象。研究发现在振子数 N ≤ 3 情形下系统不会出现多态现象，而 N ≥ 4 的多振子系统则呈现有规律的多同步定态。我们进一步对耦合振子系统中出现的多定态规律及定态稳定性进行了理论分析，得到定态下的渐近稳定解。数值模拟得出的多体系统同步区特征和理论描述相一致。研究结果显示在绝热条件下随着耦合强度的减小，系统从不同分支的同步态出发最终会回到同一非同步态。这说明，耦合振子系统在非同步区由于运动的遍历性只具有单一的非同步态，而在发生同步时由于遍历性破缺会产生多个同步定态的共存现象。第五章研究一维单向耦合振子链的同步动力学演化机制。我们根据平均频率分岔树得出非同步区由于振子频率的差异，振子之间存在多种不同的同步方式。为了解释产生这种同步现象的背后机制，我们理论研究了 N = 3 的少体系统及该系统中各种不同方式的同步竞争机制。在此少体系统分析方法的基础上，我们将此结论推广到一般的多振子耦合链，发现理论分析结果和数值模拟结果吻合。本章中关于耦合系统中振子不同方式同步的分析对复杂网络集团化现象的研究具有一定的启示作用。第六章为全文总结。

Spontaneous organization and its collective behaviors of nonlinear coupling systems is one ofthe most important topics in the current study of nonlinear science. After being familiar with thechaos or non-chaos behaviors and disciplines on single nonlinear system, researchers in this fieldhave moved their interests to the coupling nonlinear systems since eighty’s of last century. Theirresearch works mainly focus on the generation mechanism on all kinds of collective behaviors ofself-organization systems and their mutual transformations. These behaviors include synchronization, phase-locking, frequency-locking, the death of vibration, nonlinear wave, extending state, theanti-phase state and so on. In this paper, we primarily discuss the collective behavior of synchronization, frequency-locking, phase-locking problems in the nonlinear coupling systems. We studythe coupling system with Kuramoto oscillators and adopt the combination of numerical simulationand theoretical analysis methods in this paper. The phenomenon of cooperation and dynamic behaviors of different Kuramoto oscillator systems are discussed when the complete synchronizationstate, partial synchronization or phase synchronization state is obtained.This thesis is arranged as following:In chapter 1, we briefly introduce the basic background knowledge of nonlinear dynamicssystem, which includes the expressions of nonlinear system dynamics equation, the solution ofthe linear stability analysis, the type of fixed-point and the bifurcation. We consider the commondynamic equation describing the nonlinear coupled system, namely the Kuramoto model. The average field solution of this model with the all-to-all global coupling and related research methodsof local coupling systems are provided in this chapter. We also give the basic concept of commoncomplex network, the network types, the decision method of network synchronization and synchronization analysis on complex network with Kuramoto oscillators. All these basic knowledgelay the foundation for our understanding on the discussion followed.In chapter 2, we focus on a particular type of synchronization in a ring of Kuramoto oscillators, single-clustering synchronization (SCS). It is found that SCS occurs for a particular spatialfrequency distribution and gives rise to a very small critical coupling strength Kc even if the oscillator number is large. We provide some solid evidence for the convergence of Kc to a smallconstant in the thermodynamic limit, based on the finite size analysis. Furthermore, we demonstrate that it is robust in the sense of either switching the natural frequencies of any two oscillatorsor randomly perturbing the frequencies of all coupled oscillators. All these findings suggest thatthe single-clustering synchronization is indeed generically observable, with merit for potentialengineering applications.In chapter 3, we deal with the emergence of explosive synchronization in scale-free networksby considering the Kuramoto model of coupled phase oscillators. The explosive synchronizationcan be produced by the correlation between natural frequencies of oscillators and their degrees,and we include the frustration effect in the systems to discuss the phase synchronization. Ourresults show that the frustration can enhance or delay the transition to reach a synchronous state,the desynchronization can be obtained with the change of the frustration value and the continuity ofphase transition also be modified. We study the star network with typical scale-free properties, ourtheoretical results are consistent with the numerical simulations. We also propose a self-consistentmethod to understand the desynchronization with the increase of the frustration. Our findings haveimportant implications in controlling synchronization in complex networks since the frustration isa continuous controllable parameter in experiment, and a discontinuous abrupt in engineering isalways dangerous.In chapter 4, the dynamics of a ring of Kuramoto phase oscillators coupled with unidirectionalcouplings is investigated. Multiple collective states are observed in the synchronization regionwith the increase of the coupling strength for N > 4 , which cannot be found for N ≤ 3. We furtherpresent theoretical analysis on the feature and stability of the multiple synchronous states andobtain the asymptotically stable solutions. The characteristic in synchronization region computedby numerical simulation coincides with theoretical calculation very well. The systems for whichoriginal states belong to different stable states will evolve to the same incoherent state with theadiabatic decrease of coupling strength, suggesting that the only one incoherent state is attributedto the ergodicity of the phase space of coupled oscillators in asynchronous region. When thesystem reaches synchronization, the phenomenon of the coexistence of multiple stable states willemerge because of the broken ergodicity. All these analyses indicate that the multiple states ofsynchronization are indeed generically observable.In chapter 5, we discuss the synchronous dynamic evolution mechanism of unidirectionalcoupled oscillators in one-dimensional chain. According to the bifurcation tree of oscillators’ frequency, we observe a variety of synchronization styles for the oscillators with different frequenciesin small coupling region. In order to understand this potential synchronization mechanism, we analyze the N = 3 system and its competition mechanism of different types of synchronization.Based on the results of the N = 3 system, we extend the conclusion to the general systems withmany oscillators in a chain, theoretical results coincide with numerical simulations. Our resultshas certain guiding significance on the group synchronization study of complex networks.The summary of the whole article is presented in the last chapter.